In this work, the collocation method to solve fuzzy Volterra integral equations is introduced. Then, solving these systems by collocation method with Bernoulli polynomials is presented. Also, we will discuss the convergence of this method. In addition, the superiority of this method compared to other methods is proved, by solving of some systems of fuzzy Volterra integral equations.
AbbasbandyS., BabolianE. and AlaviM., Numerical method for solving linear Fredholm fuzzy integral equations ofthe second kind, Chaos Solitons Fract31(1) (2007)138–146.
2.
AllahviranlooT. and BehzadiS.S., The use of airfoil and Chebyshev polynomials methods for solving fuzzyFredholm integro-differential equations with Cauchy kernel, Soft Computing (2014)1885–1897.
3.
AllahviranlooT., SalehiP. and NejatiyanM., Existence and uniqueness of the solution of nonlinear fuzzyVolterra integral equations, Iranian Journal of Fuzzy Systems2 (2015)75–86.
AnastassiouG.A. and GalS.G., On a fuzzy trigonometric approximation theorem of Weirstrass-type, J FuzzyMath9(3) (2001)701–708.
6.
AttariH. and YazdaniA., A computational method for fuzzy Volterra-Fredholm integral equations, FuzzyInform Eng2 (2011)147–156.
7.
BalachandranK. and KanagarajanK., Existence of solutions of general nonlinear fuzzy Volterra-Fredholm integralequations, J Appl Math Stochastic Anal3 (2005)333–343.
8.
BalachandranK. and PrakashP., On fuzzy Volterra integral equations with deviating arguments, J Appl MathStochastic Anal2 (2004)169–176.
9.
BalachandranK. and PrakashP., Existence of solutions of nonlinear fuzzy Volterra-Fredholm integral equations, J Pure Appl Math33 (2002)329–343.
10.
BarkhordariM., Ahmadi and M. Khezerloo, Fuzzy bivariate Chebyshev method for solving fuzzy Volterra-Fredholmintegral equations, Int J Indust Math3(2) (2011)67–77.
11.
BehzadiS.S., AllahviranlooT. and AbbasbandyS., Solving fuzzy second-order nonlinear Volterra-Fredholmintegro-differential equations by using Picard method, J Neural Computing Appl (2012)337–346.
12.
BehzadiS.S., AllahviranlooT. and AbbasbandyS., The use of fuzzy expansion method for solving fuzzy linearvolterra-fredholm integral equations, Journal of Fuzzy and Intelligent Systems4 (2014)1817–1822.
13.
BicaA.M., Error estimation in the approximation of the solutions of nonlinear fuzzy Fredholm integral equations, Info Sci178 (2008)1279–1292.
14.
BicaA.M., One-sided fuzzy numbers and applications to integral equations from epidemiology, Fuzzy SetsSyst219 (2013)27–48.
15.
BrunnerH., On the numerical solution of nonlinear Volterra-Fredholm integral equations by collocation method, SIAM J Numer Anal27(4) (1990)987–1000.
16.
ChangS.S.L. and ZadehL., On fuzzy mapping and control, IEEE Trans Syst Man Cybernet2 (1972)30–34.
17.
DuboisD. and PradeH., Operations on fuzzy numbers, J Syst Sci9 (1978)613–626.
18.
DuboisD. and PradeH., Fuzzy sets and systems, Theory and Applications, 1980–New YorkAcademic Press.
19.
EzzatiR., A method for solving dual fuzzy general linear systems, Appl Comput Math7 (2008)235–241.
20.
EzzatiR. and ZiariS., Numerical solution and error estimation of fuzzy Fredholm integral equations using fuzzystein polynomials, Aust J Basic Appl Sci5(9) (2011)2072–2082Bern.
21.
FariborziM.A., Araghi and N. Parandin, Numerical solution of fuzzy Fredholm integral equations by the Lagrangeinterpolation based on the extension principle, Soft Comput15 (2011)2449–2456.
22.
GoetschelR. and VoxmanW., Elementary fuzzy calculus, Fuzzy Sets Syst18 (1986)31–43.
KauthenJ.P., Continuous time collocation method for Volterra-Fredholm integral equations, Numer Math56(5) (1989)409–424.
25.
KhezerlooM., AllahviranlooT., SalahshourS. and KhorasaniM., Kiasari and S. Haji Ghai, Application of Gaussianquadratures in solving fuzzy Fredholm integral equations, Information Processing and Management of Uncertainty inKnowledge- Based Systems, Applications, Communications in Computer and Information Science81 (2010)481–490sem.
26.
KhorasaniS.M., Kiasari, M. Khezerloo and M.H. Dogani Aghcheghloo, Numerical solution of linear Fredholm fuzzyintegral equations by modified homotopy perturbation method, Aust J Basic Appl Sci4 (2010)6416–6423.
27.
KlirG.J., ClairU.S. and YuanB., Fuzzy set theory, foundations and applications, Inc, 1997–Prentice-Hall.
28.
MaM., FriedmanM. and KandelA., A new fuzzy arithmetic, Fuzzy Sets Syst108 (1999)83–90.
29.
MirzaeeF., ParipourM. and KomakM., Yari, Numerical solution of Fredholm fuzzy integral equations of the secondkind via direct method using triangular functions, Journal of Hyperstructures1(2) (2012)46–60.
30.
MirzaeeF., ParipourM. and KomakM., Yari, Application of hat functions to solve linear Fredholm fuzzy integralequation of the second kind, J Intell Fuzzy Syst27(1) (2014)211–220.
31.
MirzaeeF. and KomakM., Yari and M. Paripour, Solving linear and nonlinear Abel fuzzy integral equations by homotopyanalysis method, Journal of Taibah University for Science9(1) (2015)104–115.
32.
MizumotoM. and TanakaK., The four operations of arithmetic on fuzzy numbers, Syst Comput Controls7(5) (1976)73–81.
33.
MizumotoM. and TanakaK., Some propertise of fuzzy numbers, Advances in Fuzzy Set Theory and Applications,NorthHolland, pp, (1979)153–164Amsterdam.
34.
MolabahramiA., ShidfarA. and GhyasiA., An analytical method for solving linear Fredholm fuzzy integralequations of he second kind, Comput Math Appl61(211), 2754–2761.
NandaS., On integration of fuzzy mappings, Fuzzy Sets Syst32 (1989)95–101.
37.
OrdokhaniY., An application of Walsh functions for Fredholm-Hammerstein integro-differential equations, IntJ Contemp Math Sci5(22) (2010)1055–1063.
38.
OtadiM. and MoslehM., Numerical solution of fuzzy integral equations using stein polynomials, Aust JBasic Appl Sci5(7) (2011)724–728Bern.
39.
ParandinN. and FariborziM.A., Araghi, The approximate solution of linear fuzzy Fredholm integral equations ofthe second kind by using iterative interpolation, Proc World Acad Sci Eng Technol37 (2009)1036–1042.
40.
ParandinN. and FariborziM.A., Araghi, The numerical solution of linear fuzzy Fredholm integral equations of thesecond kind by using finite and divided differences method, Soft Comput15 (2010)729–741.
41.
ParipourM. and KomakM., Yari, Existence and uniqueness of solutions for Fuzzy quadratic integral equation offractional order, J Intell Fuzzy Syst32(3) (2017)2327–2338.
42.
PuriM.L. and RalescuD., Fuzzy random variables, J Math Anal Appl114(2) (1986)40–94.
43.
PuriM.L. and RalescuD., Differentials of fuzzy functions, J Math Anal Appl91 (1983)552–558.
44.
SadeghiH., Goghary and M. Sadeghi Goghary, Two computational methods for solving linear Fredholm fuzzy integralequation of the second kind, Appl Math Comput182 (2006)791–796.
45.
SalahshourS. and AllahviranlooT., Application of fuzzy differential transform method for solving fuzzy Volterraintegral equations, Appl Math Model37(3) (2013)1016–1027.
46.
SalehiP. and NejatiyanM., Numerical method for nonlinear fuzzy Volterra integral equations of the second kind, Int J Indust Math3(3) (2011)169–179.
47.
SeikkalaS., On the fuzzy initial value problem, Fuzzy Sets Syst24 (1987)319–330.
48.
ShafieeM., AbbasbandyS. and AllahviranlooT., Predictor correctormethod for nonlinear fuzzy Volterra integral equations, Aust JBasic Appl Sci2(12) (2011)2865–2874.
49.
TohidiE., BhrawyA.H. and ErfaniK., A collocation method based on oulli operational matrix for numericalsolution of generalized pantograph equation, Appl Math Model37 (2013)4283–4294Bern.
50.
WatkinsD.S., Fundamentals of matrix computations, John Wiley and Sons64 (2004).
51.
WuH.C., The improper fuzzy Riemann integral and its numerical integration, Inform Sci111 (1998)109–137.
52.
WuH.C., The fuzzy Riemann integral and its numerical integration, Fuzzy Sets Syst110 (2000)1–25.
53.
ZadehL.A., Linguistic vriables, approximate reasoning and disposition, J Med Inform8 (1983)173–186.
54.
ZadehL.A., The concept of a linguistic variable and its application to approximate reasoning, Inform Sci8 (1975)199–249.
